Functions and graphs

# Other characteristics

A function is called **increasing** in an interval if, for all x_{1} and x_{2} in the interval such that x_{1} < x_{2}, then f(x_{1}) < f(x_{2}).

A function is called **decreasing** in an interval if, for all x_{1} and x_{2} in the interval such that x_{1} < x_{2}, then f(x_{1}) > f(x_{2}).

The **maximum** and **minimum** of a function, known collectively as **extrema**, are the largest and smallest value that the function takes at a point either within a given neighbourhood (**local or relative extremum**) or on the function domain in its entirety (**global or absolute extremum**).

Examples:

**Exercise**: find the extrema, increasing and decreasing intervals of these functions:

a)

b)

Solutions:

function | increasing | decreasing | rel max | rel min | abs max | abs min |

a) | (0,∞) | (-∞,0) | Φ | 0 | Φ | 0 |

b) | (-∞,-3.5)U(-1.5,1)U(1,2) | (-3.5,-1.5)U(2,∞) | -3.5,2 | -1,5 | 2 | Φ |

Licensed under the Creative Commons Attribution Non-commercial Share Alike 3.0 License